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Your data matches 354 different statistics following compositions of up to 3 maps.
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Matching statistic: St000058
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 4
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 4
[4,3,2,1] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,2] => [1,1]
=> 1
[2,1] => [2]
=> 2
[1,2,3] => [1,1,1]
=> 1
[1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1]
=> 2
[2,3,1] => [3]
=> 3
[3,1,2] => [3]
=> 3
[3,2,1] => [2,1]
=> 2
[1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [3,1]
=> 3
[1,4,2,3] => [3,1]
=> 3
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 3
[2,3,4,1] => [4]
=> 4
[2,4,1,3] => [4]
=> 4
[2,4,3,1] => [3,1]
=> 3
[3,1,2,4] => [3,1]
=> 3
[3,1,4,2] => [4]
=> 4
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [3,1]
=> 3
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [4]
=> 4
[4,1,2,3] => [4]
=> 4
[4,1,3,2] => [3,1]
=> 3
[4,2,1,3] => [3,1]
=> 3
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [4]
=> 4
[4,3,2,1] => [2,2]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St001555
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001555: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 3
[3,1,2] => [3,1,2] => 3
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => 3
[1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 3
[2,3,4,1] => [2,3,4,1] => 4
[2,4,1,3] => [2,4,1,3] => 4
[2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => [3,1,2,4] => 3
[3,1,4,2] => [3,1,4,2] => 4
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 4
[4,1,2,3] => [4,1,2,3] => 4
[4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [4,3,1,2] => 4
[4,3,2,1] => [4,3,2,1] => 2
Description
The order of a signed permutation.
Matching statistic: St000141
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [1,3,2] => 1 = 2 - 1
[3,1,2] => [3,1,2] => 2 = 3 - 1
[3,2,1] => [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,2,4,3] => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [1,3,2,4] => 1 = 2 - 1
[2,3,4,1] => [1,2,4,3] => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => [1,4,3,2] => 2 = 3 - 1
[3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => [2,1,4,3] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => [2,1,4,3] => 1 = 2 - 1
[3,4,1,2] => [2,4,1,3] => 2 = 3 - 1
[3,4,2,1] => [1,4,3,2] => 2 = 3 - 1
[4,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => [4,1,3,2] => 3 = 4 - 1
[4,2,1,3] => [4,2,1,3] => 3 = 4 - 1
[4,2,3,1] => [4,1,3,2] => 3 = 4 - 1
[4,3,1,2] => [4,3,1,2] => 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000319
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 1 - 1
[1,2] => [1,1]
=> 0 = 1 - 1
[2,1] => [2]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> 1 = 2 - 1
[2,3,1] => [3]
=> 2 = 3 - 1
[3,1,2] => [3]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> 1 = 2 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 1 - 1
[1,2] => [1,1]
=> 0 = 1 - 1
[2,1] => [2]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> 1 = 2 - 1
[2,3,1] => [3]
=> 2 = 3 - 1
[3,1,2] => [3]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> 1 = 2 - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
Mp00108: Permutations —cycle type⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 1 - 1
[1,2] => [1,1]
=> 0 = 1 - 1
[2,1] => [2]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> 1 = 2 - 1
[2,3,1] => [3]
=> 2 = 3 - 1
[3,1,2] => [3]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> 1 = 2 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1
[1,2] => [1,1]
=> [2]
=> 1
[2,1] => [2]
=> [1,1]
=> 2
[1,2,3] => [1,1,1]
=> [3]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> 2
[2,1,3] => [2,1]
=> [2,1]
=> 2
[2,3,1] => [3]
=> [1,1,1]
=> 3
[3,1,2] => [3]
=> [1,1,1]
=> 3
[3,2,1] => [2,1]
=> [2,1]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 3
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 3
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 2
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 2
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 3
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 3
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 3
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 2
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 3
[3,4,1,2] => [2,2]
=> [2,2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 3
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 3
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 2
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 4
[4,3,2,1] => [2,2]
=> [2,2]
=> 2
Description
The length of the partition.
Matching statistic: St000013
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[2,4,3,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000015
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
Description
The number of peaks of a Dyck path.
The following 344 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000240The number of indices that are not small excedances. St000308The height of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000638The number of up-down runs of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001497The position of the largest weak excedence of a permutation. St001530The depth of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000651The maximal size of a rise in a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000005The bounce statistic of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000390The number of runs of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000628The balance of a binary word. St000653The last descent of a permutation. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000833The comajor index of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000912The number of maximal antichains in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001343The dimension of the reduced incidence algebra of a poset. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001725The harmonious chromatic number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St000012The area of a Dyck path. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000080The rank of the poset. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000439The position of the first down step of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000548The number of different non-empty partial sums of an integer partition. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000742The number of big ascents of a permutation after prepending zero. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000983The length of the longest alternating subword. St000989The number of final rises of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001375The pancake length of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001864The number of excedances of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001062The maximal size of a block of a set partition. St000956The maximal displacement of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000844The size of the largest block in the direct sum decomposition of a permutation. St000503The maximal difference between two elements in a common block. St000730The maximal arc length of a set partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000504The cardinality of the first block of a set partition. St000675The number of centered multitunnels of a Dyck path. St000823The number of unsplittable factors of the set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000925The number of topologically connected components of a set partition. St001346The number of parking functions that give the same permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001721The degree of a binary word. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000477The weight of a partition according to Alladi. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. St001389The number of partitions of the same length below the given integer partition. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000997The even-odd crank of an integer partition. St001060The distinguishing index of a graph. St001571The Cartan determinant of the integer partition. St001645The pebbling number of a connected graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000438The position of the last up step in a Dyck path. St000981The length of the longest zigzag subpath. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000035The number of left outer peaks of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001875The number of simple modules with projective dimension at most 1. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000806The semiperimeter of the associated bargraph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000366The number of double descents of a permutation. St001626The number of maximal proper sublattices of a lattice. St000264The girth of a graph, which is not a tree. St000307The number of rowmotion orbits of a poset. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000254The nesting number of a set partition. St000502The number of successions of a set partitions. St001061The number of indices that are both descents and recoils of a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001737The number of descents of type 2 in a permutation. St001863The number of weak excedances of a signed permutation. St000023The number of inner peaks of a permutation. St000174The flush statistic of a semistandard tableau. St000353The number of inner valleys of a permutation. St000422The energy of a graph, if it is integral. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001469The holeyness of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St001568The smallest positive integer that does not appear twice in the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation.
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